3 Answers
3

If you're willing to allow some high-power Riemannian geometry theorems, the classification of constant curvature surfaces tells you that your surface is isometric to a sphere, the Euclidean plane, or the hyperbolic plane. It can then be checked by computation that all geodesic circles are constant-curvature.

Consider the three cases K=0, K>0 and K<0 in your second equation. In each case, you can use ODE theory (and the values of E, F and limits of G) to solve for G, which in all cases is independent of theta. Plug in the first fundamental form coefficients into your first equation for the geodesic curvature.